| L(s) = 1 | − 9.41e10·3-s + 8.47e18·7-s + 8.86e21·9-s − 7.81e25·13-s + 5.14e29·19-s − 7.97e29·21-s + 1.42e32·25-s − 8.34e32·27-s + 1.47e34·31-s − 2.15e36·37-s + 7.35e36·39-s − 2.42e37·43-s − 6.77e38·49-s − 4.84e40·57-s − 1.52e41·61-s + 7.50e40·63-s − 1.85e42·67-s + 1.27e43·73-s − 1.33e43·75-s + 4.66e43·79-s + 7.85e43·81-s − 6.61e44·91-s − 1.38e45·93-s − 2.33e45·97-s − 1.38e45·103-s + 4.28e46·109-s + 2.02e47·111-s + ⋯ |
| L(s) = 1 | − 3-s + 0.309·7-s + 9-s − 1.87·13-s + 1.99·19-s − 0.309·21-s + 25-s − 27-s + 0.736·31-s − 1.83·37-s + 1.87·39-s − 0.654·43-s − 0.904·49-s − 1.99·57-s − 1.31·61-s + 0.309·63-s − 1.85·67-s + 1.77·73-s − 75-s + 1.05·79-s + 81-s − 0.579·91-s − 0.736·93-s − 0.469·97-s − 0.0703·103-s + 0.590·109-s + 1.83·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(47-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+23) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{47}{2})\) |
\(\approx\) |
\(1.242341305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.242341305\) |
| \(L(24)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{23} T \) |
| good | 5 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 7 | \( 1 - 8472089095515545378 T + p^{46} T^{2} \) |
| 11 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 13 | \( 1 + \)\(78\!\cdots\!58\)\( T + p^{46} T^{2} \) |
| 17 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 19 | \( 1 - \)\(51\!\cdots\!06\)\( T + p^{46} T^{2} \) |
| 23 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 29 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 31 | \( 1 - \)\(14\!\cdots\!54\)\( T + p^{46} T^{2} \) |
| 37 | \( 1 + \)\(21\!\cdots\!06\)\( T + p^{46} T^{2} \) |
| 41 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 43 | \( 1 + \)\(24\!\cdots\!38\)\( T + p^{46} T^{2} \) |
| 47 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 53 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 59 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 61 | \( 1 + \)\(15\!\cdots\!86\)\( T + p^{46} T^{2} \) |
| 67 | \( 1 + \)\(18\!\cdots\!22\)\( T + p^{46} T^{2} \) |
| 71 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 73 | \( 1 - \)\(12\!\cdots\!66\)\( T + p^{46} T^{2} \) |
| 79 | \( 1 - \)\(46\!\cdots\!78\)\( T + p^{46} T^{2} \) |
| 83 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 89 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 97 | \( 1 + \)\(23\!\cdots\!62\)\( T + p^{46} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.73372068677604117683102552609, −10.41369303094410391209993234242, −9.462307269118393366792703855653, −7.66527887760274664647297891946, −6.80244896655980927178388595115, −5.29989876522081678604714347235, −4.75046018657932182929276075097, −3.09150532796014429931367731081, −1.64517794259348407465409983657, −0.52625139447811575774616068696,
0.52625139447811575774616068696, 1.64517794259348407465409983657, 3.09150532796014429931367731081, 4.75046018657932182929276075097, 5.29989876522081678604714347235, 6.80244896655980927178388595115, 7.66527887760274664647297891946, 9.462307269118393366792703855653, 10.41369303094410391209993234242, 11.73372068677604117683102552609
